Colonization and Collapse on Homogeneous Trees

The Maki–Thompson rumor model is defined by assuming that a population represented by a graph is subdivided into three classes of individuals; namely, ignorants, spreaders and stiflers. A spreader tells the rumor to any of its nearest ignorant neighbors at rate one. At the same rate, a spreader becomes a stifler after a contact with other nearest neighbor spreaders, or stiflers. In this work we study the model on random trees. As usual we define a critical parameter of the model as the critical value around which the rumor either becomes extinct almost-surely or survives with positive probability. We analyze the existence of phase-transition regarding the survival of the rumor, and we obtain estimates for the mean range of the rumor. The applicability of our results is illustrated with examples on random trees generated from some well-known discrete distributions.

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“…See equation (3.1) in [27] for more details. Therefore, by applying proposition 4.2 in [20] we get lim n→∞ θ n (ν) = θ(ν).…”

Section: The Model On a Random Tree And First Resultsmentioning

confidence: 91%

Stochastic rumors on random trees

Valdivino¹,

Rodríguez²,

Speroto³

2021

J. Stat. Mech.

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“…Artalejo et al [1] and Brockwell [2,3] analyzed models for populations that after catastrophes, the survivors individuals remain together in the same colony. Schinazi [15], Machado et al [11] and Junior et al [7,8,10] studied models for populations that after catastrophes, individuals disperse trying to make new colonies to improve the odds of their species survival. In all these works, different types of catastrophes and different dispersion strategies were considered.…”

Section: Introductionmentioning

confidence: 99%

Extinction time in growth models subject to geometric catastrophes

Vargas¹,

Machado²,

Roldán-Correa³

2023

J. Stat. Mech.

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Recently, different dispersion strategies in population models subject to geometric catastrophes have been considered as strategies to improve the chance of population’s survival. Such dispersion strategies have been contrasted with the strategy where there is no dispersion, comparing the probabilities of survival. In this article, we contrast survival strategies when extinction occurs almost surely, evaluating which strategy prolongs population’s life span. Our results allow one to analyze what is the best strategy based on parameters as the probability that each individual exposed to catastrophe survives, the growth rate of the colony, the type of dispersion and the spatial restrictions.

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“…These models are of interest for gaining a deeper understanding of quantities such as population survival probability, extinction time distribution, mean number of individuals removed and the distribution of maximum population size. Previous studies [1, 5-7, 14, 17, 18] pertain to population models where catastrophe survivors remain united in the same colony, whereas the models examined in [11,12,19,20,22] investigate population dynamics with survivors dispersing to establish new colonies elsewhere. In these papers, different types of catastrophes and different dispersion schemes are considered to analyze whether combining some of these schemes increases population viability.…”

Section: Introductionmentioning

confidence: 99%

“…The models analyzed in [11,12,19,20,22] aim to establish the best strategy, i.e. dispersion or no dispersion, based on the survival probability of the population.…”

Section: Introductionmentioning

confidence: 99%

Extinction time in growth models subject to binomial catastrophes ^*

Duque,

Junior,

Machado

et al. 2023

J. Stat. Mech.

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Populations are often subject to catastrophes that lead to significant reductions in the number of individuals. Many stochastic growth models have been considered to explain such dynamics. Among the reported results, it has been considered whether dispersion strategies, at times of catastrophes, increase the survival probability of the population. In this paper, we contrast dispersion strategies by comparing the mean extinction times of a population under conditions of near-certain extinction. Specifically, we consider populations subject to binomial catastrophes, where the population size is reduced according to a binomial law when a catastrophe occurs. Our findings delineate the optimal strategy (dispersion or non-dispersion) based on variations in model parameter values.

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Colonization and Collapse on Homogeneous Trees

Cited by 4 publications

References 9 publications

Stochastic rumors on random trees

Stochastic rumors on random trees

Extinction time in growth models subject to geometric catastrophes

Extinction time in growth models subject to binomial catastrophes ^*

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Colonization and Collapse on Homogeneous Trees

Cited by 4 publications

References 9 publications

Stochastic rumors on random trees

Stochastic rumors on random trees

Extinction time in growth models subject to geometric catastrophes

Extinction time in growth models subject to binomial catastrophes *

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Extinction time in growth models subject to binomial catastrophes ^*